Like @tmgarcia86 said (thanks!), this is not the verb fazer, but the verb ficar. In the example you gave, it is used to emphasize that there was a change in her state of mind compared to how she felt before. She became furious, she wasn’t furious before.
When you say “Ela está furiosa”, all you’re saying is that she is currently furious, but maybe she was already furious before. It is not clear if her fury is a whole new emotion or not.
So, in this context, estar and ficar don’t really overlap in terms of meaning. On the other hand, if you’re describing a position or a time period, the two verbs may at times be interchangeable. For example:
A estação está no centro da cidade. = A estação fica no centro da cidade. (The station is in the city center.)
@stephencanthony You’re right though that fazer is a tricky one. These are some of the idiomatic uses that I found in one of my grammar books:
fazer perguntas - to ask questions
fazer compras - to shop
fazer uma visita - to pay a visit
fazer uma viagem - to take a trip
fazer um passeio - to take a walk / to go for a ride
fazer a barba - to shave
fazer falta - to be needed / to be lacking
não faz mal - it’s all right / it doesn’t matter
faz frio - it is cold (Used to talk about weather)
We use the verb Ficar in situations that describe a sudden change. You can use it with everything and we use it in the Past Tense (PPS Pretérito Perfeito Simples) and Presente Tense (Presente do Indicativo).
Phrases like “Ele ficou rico” (He got rich, meaning that he was never rich) or “Ele ficou gordo” (He got fat, with the same meaning as before) and as @Joseph said changes in the state of mind.
With the Presente Tense, it’s normally used only with changes in the state of mind.
Ela fica feliz quando come comida portuguesa. (She’s happy when she eats portuguese food.)
I have a list of really disliked verbs/terms in Portuguese. Fazer is one of them. Now I get to add ficar to my list!. To really understand the difference is hard for me. I am forced to deal with these two verbs from time to time, but I am never sure just exactly what they mean, Deixar is another on my list of non understandable verbs. Dizer has given me some issues, but I think that one is resolved now. Achar is not popular with me either!
@hscharnhorst, haha! All those verbs you mentioned are very important, though, so I’m sure you’ll be spending quite some time with them. Maybe this is what people mean by “keep your friends close and your enemies closer”